The conclusion of World War I did not bode well for
Germany. The nation went into a state of economic depression. This was
caused by the Versailles Treaty as it put limits on Germany's industrial
production, hence severely limiting economic growth. Due to the limited
amount of employment, national moral plummeted. This was the spring board
for Adolf Hitler's rise in power.

In defiance of the Versailles Treaty of 1919, Germany
was strengthening her military forces from year to year. Abroad, intelligence
operations were being conducted by Germany's numerous intelligence offices,
where the military was disguised as Abwehr, [or military intelligence],
espionage - near the borders with Poland and Czechoslovakia. Germany had
also commissioned a radio intelligence organization that employed many
monitoring personnel and decryptment specialists. Unfortunately, little
is known about the first phase in the Polish-German cryptological duel,
which all started immediately following World War I, as well as the reemergence
of Poland as an independent state.

In the middle 1920s, German radio intelligence started
reading messages between the staffs of the Polish Air Force. Accurate information
was gained about locations of most of the units. Poland's Section II (military
intelligence) had noticed a setback; for some period of time, the instructions
for German agent in Poland hardly carried any assignments concerning air
force units, whereas the Polish Air Force and air defenses had been really
interesting to German intelligence. This all suggested that information
was leaking somehow, and assignments had to be shifted. The Polish decided
to dole out some false information in this same fashion to throw the Germans
off course, while still communicating with themselves through courier and
coded telephone messages. The Abwehr never detected this code, entitled
"Port," and this more or less neutralized the damage the Germans had previously
caused.

In the later half of 1932, Polish-German relations
were exceptionally strained. For many years Germany had been waging tariff
and economic war against Poland. Radio monitoring allowed for Poland to
learn of Germany's diplomatic and military plans, provided that there were
ways developed for solving German codes and ciphers. This was becoming
harder all the time, though, because Germany was increasingly using machine
ciphers. Every so often, the Germans tried to make things a little more
difficult. They altered from alphabetical order, to leaving out letters
of the alphabet, to even broadcasting bogus code groupings.

The earliest Polish work on the German machine cipher
had begun in 1927-1928, a short while after the system had been introduced
by the German Navy and Army. Germany's armed forces faced the dilemma of
"reconciling the requirements of security with the imperatives of speed
and convenience" [Kozaczuk]. Since 1918 the German Navy had been contemplating
the use of cipher machines. Many designs were considered, but there was
one developed by Dr. Arthur Scherbius, that became very powerful. The name
of the machine was the "Enigma." This machine was then patented. Attempts
had been made to solve Enigma by resorting to leading mathematicians and
to parapsychology.

Other than mathematical analysis, there was nowhere
else to start in deciphering this code. Clues had to come from the cipher
texts, which were meaningless sequences of letters. The frequencies of
the letters was almost uniform, hence using statistics to break this code
was useless. "The work of Enigma required enormous concentration and at
least eighty intercepts collected on the same day, using the same setting
on the German cipher devices" [Kozaczuk].

On the outside,
the machine resembled a typewriter, with an extra panel built into the
lid. There were twenty-six little circular glass windows in the panel,
which bared the letters of the alphabet, just like the keyboard. On the
underside of the panel were a corresponding number of glowlamps. Inside
the machine there was originally a set of three rotors, otherwise known
as rotating drums, and a "reversing drum" all mounted on one axle. An intricate
system of wiring included this axle. Powering this machine was done by
a battery or a regular current which passed through a small transformer.

When a key was pressed, one or more of the rotors
revolved. Concurrently, a glowlamp beneath the panel lit up, illuminating
the letter in the window above it. The design of the Enigma machine allowed
the user to "type" the plain text while the letters of the cipher text
lit in the appropriate windows and, conversely, when one "typed" out a
cipher text, the letter illumination spelled out the plain text. The secret
dialog was only conducted through both parties having the identical devices
set, using various knobs and levers, to the same cipher key. Although the
commercial Enigma gave general insight to the machine's construction and
operation, the military version had completely different wiring and additional
components.

Enigma's breaking was perchance the most sensational
event, in terms of difficulty and of consequences, in the whole several-thousand-year
history of cryptography, or ("secret writing"), and cryptology, or (the
study of secret writing, chiefly for reasons of decryptment - the "breaking,"
or "reading," of secret correspondence by a third party.) Cryptology is
oriented around ciphers, which shift, or transpose, letters or substitute
them for original letters which appear in a message. Cryptology also deals
with codes that have substitution of arbitrary symbols consisting of letters
or numbers for entire words and phrases.

It all began in ancient Greece. The roots of cryptography
and cryptology began to grow. The history of this field contains names
such as Julius Caesar, who invented "Caesar ciphers." These codes deal
with the shifting of letters to a specific key letter. Another famous name
is Blaise de Vigenere, where his coding system revolves around a table
of all the two-letter combinations of the alphabet. Thomas Jefferson invented
the "wheel cypher." Edgar Allen Poe demonstrated that any substitution
cipher can be solved given long enough cipher text. In the computer world,
we have Charles Babbage who first attempted to apply algebra to the solution
of ciphers. A noted linguist named August Kerckhoffs codified knowledge
about ciphers and determined that the variability of the key determines
the strength of the cipher rather than the physical security of the device.
"Cryptography and cryptology have developed in parallel with the evolution
of science and technology" [Kozaczuk].

The cryptologists would need to continue studying
the Enigma system from the mathematical side in order to break the code.
Group theory, one of the divisions of higher mathematics, and the properties
of permutation groups would be very useful tools in studying the military
Enigma.

Since 1927, the Polish Cipher Bureau had been working
on the Enigma cipher. Other than intercepts, there was nothing else that
the Polish had to go on, so any new information was welcome. France and
Czechoslovakia, like Poland, were threatened by German expansionism. They
were also allies for Poland in collecting intelligence on German armaments
and plans for war. Even though France and Poland had their disagreements
since their alliance in February of 1921, cooperation among their intelligence
services was very important.

In December of 1932, tasks were established between
the nations. The French were to put their concentration towards furnishing
intelligence from Germany which might facilitate the breaking of the machine
cipher, while the Poles worked with theoretical studies of Enigma intercepts.
German radio intercepts were to be exchanged while radiogonometric data
was to be studied. Other intelligence was also to be worked with during
the procedures. The Warsaw Cipher Bureau was also hiring mathematicians
who were experienced with German ciphers. "The principle of strict secrecy
and compartmentalization meant that even the most trusted radio intelligence
and encryptment workers learned only as much as was indispensable to their
own work about the materials supplied to them" [Kozaczuk]. The brains of
the German Cipher office, the three mathematician- cryptologists the Polish
had, had no knowledge of the Polish-French contacts or of the origin of
the information which was supplied to them on the Enigma.

One Polish cryptologist by the name of Marian Rejewski
received documents concerning the machine's operation, key instructions
and obsolete tables of daily keys. Rejewski discovered that in the commercial
Enigma the letters of the alphabet were represented on the circumference
of the entry ring in the same order which they appeared on a German typewriter
keyboard. He then wondered what the order would have to do with the military
Enigma. Since both keyboards are ordered the same way, Rejewski assumed
that the entry ring was set up in the same manner. After lots of research,
he determined that since the keyboard wasn't wired to the entry drum in
keyboard sequence that maybe the wiring was in alphabetical order. By testing
this hypothesis in the manner of designating connections in a specific
rotor, he realized that the right-hand rotor turned one twenty-sixth of
a revolution each time a key was depressed.

Theoretical reconstruction involved determining the
Enigma's wiring. Cryptologists discovered how the reflector, or "reversing
drum," worked. Then, one step at a time, they reconstructed all the connections,
including a system of rotors, a common axle, and the commutator with its
plug connections. This enabled doubles of the Enigma to be built by the
Poles so they could read German ciphers. The keys had to be found, though,
prior to the encipherment of a message. By daily research through monitoring
stations, the Enigma keys were reconstructed.

February 1933 began a period of "Enigma-doubling."
Duplicates were being made of the military version of the Enigma. By 1934,
over a dozen of these machines existed. To recover the settings, or starting
positions, and the keys, the process of elimination was put into play.
The connections in the commutator would be found by using the grill method.
There were a possibility of 100,391,791,500 different connections. The
rotors had to be turned as many as 17,576 ways to find the keys. There
were 263 possible settings for each of the six possible
sequences of the three rotors. This was a time-consuming job for the mathematicians,
and, "in their haste, the men would scrape their fingers raw and bloody"
[Kozaczuk]. The Germans, being as sly as they could be, began changing
the commutator connections infrequently, once a quarter, at first. But,
in 1934, they started changing them monthly, and after a while, daily,
and to make matters even worse, they started changing the connections every
eight hours!

Rejewski then invented what he called the cyclometer.
This was a device that determined the length and number of cycles in the
"characteristic" for each position of Enigma's rotors. The cyclometer enabled
the mathematicians to set up a catalog of characteristics which could encompass
the 105,456, or 6 x 263 possible settings of the rotors.
The cyclometer also had the ability to "annihilate" the connections in
the commutator, which constituted one of Enigma's stronger points.

In November and December of 1932, the most intensive
work had been in progress on Enigma in Poland. German cipher clerks were
committing crazy blunders to try to get the Poles off their track. They
would send the same letter three times (AAA) or they would punch the letters
in alphabetical order (ABC). The Germans would also type in letters which
were in a diagonal line across the keyboard, which was also against regulations.
"Perhaps the cipher personnel were still imperfectly trained and the supervision
by their bosses superficial" [Kozaczuk]. There is a crux of the matter,
though, - blind faith in the Enigma, a belief that it could take any
message and transform it into an unbreakable cipher text. With some mathematical
analysis, Rejewski was able to reduce the number of unknowns in equations
for solving Enigma codes. So, nothing is unbreakable; with enough work,
anything can be solved.

In February of 1936, there was a special "A" key
which was introduced into secret correspondence. This is the reflector,
or reversing drum. In 1937, though, the Germans exchanged this reflector
to introduce a new "B" key. But, the Germans made a mistake; they forgot
to change the wiring in the three rotors at the same time. Because this
change was not as effective as the Germans anticipated, Enigma signals
continued to be read in Poland.

In September 1938, the Germans completely altered
the rules used by Hitler's army, navy, air force, and key civil agencies.
Rule alterations allowed for , the Enigma operator, on his own, to select
the basic position, a different one each time, for enciphering the individual
message key. But because the old keying procedures were being used, the
Polish cryptologists continued to read the trickle of information.

The cryptological difficulties continued, though. The
Polish mathematicians were thinking of constructing another device for
deciphering the messages which would be superior to the cyclometer. This
device would reduce some of the toilsome calculations. In October of 1938,
Marian Rejewski worked out the mathematical model of an aggregate which
was turned over to B.S.-4 designers at AVA Radio Manufacturing Company
in Warsaw, Poland. Jerzy Rozycki, one of the mathematicians, christened
this new machine. (This was the same company where the "doubles" were created.)
The bomb (bomba), was the new device for recovering Enigma keys. Because
every advance was a closely guarded secret, that could not be shared even
with one's family; the labor of the bomb creators had to remain anonymous.

The bomb was an electro-mechanical aggregate based
on six Polish Enigmas. Six of them were constructed at once. They were
combined with additional devices and transmissions. The bomb was composed
of electronically driven rotors which revolved automatically, creating
in each bomb, successively, 17,576 different combinations. As the rotors
aligned in the aimed-for position, a light went on, automatically the motors
stopped, and the combinations were read by the cryptologist. When the bombs
were set in motion, the daily keys could be recovered in two hours. Axle
shafts, transmission wheels, and special instantaneous glowlamps were some
of the same ideas that constituted some of the existing technology from
the Enigma. About the same time the bomb came about, there was a method
which was discovered for breaking the doubly enciphered individual message
keys to be used in accordance with the new procedures the Germans set forth
in September 1938. This new method consisted of a series of perforated
sheets of paper with the capacity of fifty-one holes by fifty-one. Each
series consisted of twenty-six sheets. Designed mainly by Henryk Zygalski,
the third mathematician, this system was effective in matching the coincident
places in this preprogrammed system. It did not matter how many plug connections
there were in the German Enigma's commutator.

Using the bombs and the perforated sheets, the cryptologists
were once again able to find keys to the signals. As soon as the key was
broken, cipher clerks set sometimes more than a dozen Enigma doubles into
operation, turning the columns into readable plain text.

In December 1938, another big change erupted. The
Germans once again revamped their Enigma ciphers. This time the change
involved more than just changing procedures. It involved changing the components
as well. The Germans introduced two additional rotors per device, making
the number of rotors five instead of three. Even though only three rotors
could be operated in any one machine at one time, the sequence that they
were used could be altered. Five rotors were selected from, which made
many more combinations possible. This, once again, nullified the Poles'
chances of decrypment by the methods so far used. But, by mathematical
and other operations during the second half of December and the first few
days of January 1939, the internal connections in rotors four and five
were reconstructed. The Poles were again on top of things and the cryptologists
had no secrets held from them. The possibilities of again reading the correspondence
of the German land and air forces opened up. Nevertheless, even though
the bomb and the perforated sheets were being used, the new keying procedure
and the increased number of rotors posed major problems. To make matters
even worse, the Germans then increased the number of plug connections in
the commutator.

So, even though the Poles continued to read the Germans'
codes without great difficulty, with all the complications added, the continuous
decrypment would have required the investment of ever greater resources.
Many cryptological bombs were in use; at least sixty were known to be existant.
Everything was getting very expensive for the Polish, as they had to train
many new skilled workers, keep up with the modern equipment, secure teletype
lines for direct transmittal of intercepts, and increase the number of
monitoring stations. Large numbers of Polish Enigmas were necessary, because
they constantly wore out from the overuse. "The solution of the Enigma
cipher just as Hitler was taking power had secured a regular flow of reliable
military, political, and other information from Germany" [Kozaczuk].

September 1939 began, and the first air raids on Warsaw
came, as well as the first dead and wounded, the first destroyed buildings,
and the aerial battles over the city. World War II started. Many were evacuated
from their stations because of the deteriorating situation. The Cipher
Bureau received orders to evacuate, along with other units of the general
staff, to Rumania. The B.S.-4 cryptologists, together with other military
personnel and civilians, crossed a bridge to Rumania on September 17, 1939.
At the border, there seemed to be some confusion. Materials were confiscated
while the military and civilians were being separated into different directions
of march. Taking advantage of all this confusion, the three mathematicians,
Rejewski, Rozycki, and Zygalski kept together, and snuck out to a railroad
station to buy tickets to Bucharest, a city located at the other end of
Rumania. They were out to contact the Polish Embassy.

Time passed, and the war continued. In 1941, a cipher
was solved that turned out to serve communication between German secret
agents. The cipher was broken by an engineer, who, while maintaining the
Enigma ciphers and radios, restored his old cryptological specialty. These
reconstructed spy reports were sent to France where the Rural Works Enterprise
discovered the transmitter and neutralized the agents. At a later time,
Rejewski, Rozycki, and Zygalski broke codes and ciphers used in German
telegraphic communications. French postal workers cooperating with the
Germans took the encrypted texts and copied them. Included were reports
from German radio-location and monitoring stations in southern France that
were used to find the secret transmitters of undergorund organizations.

The mathematicians' knowledge of the adversary increased
from knowledge of other German codes. The decryption of German messages
helped them discover the newly introduced principles of preliminary preparation
of texts and make tables of standard abbreviations, signals, and others.
In the middle of 1941, they determined that the Germans were enciphering
numbers as well, and they were varying the endings. This created more possibilities
of combinations which made decryption more difficult. The Germans had other
precautions as well. It was against the law to resend a message which had
errors in it the first time. The text had to be reedited without changing
the content. Usually, the order of the expressions would be changed.

Breaking Enigma

The mathematical solution of the Enigma cipher was put
together by Marian Rejewski. The following information appears in Appendix
E from Enigma: How the German Machine Cipher Was Broken, and How It
Was Read by the Allies in World War Two. It is a primary source by
Marian Rejewski, himself, which reports on "The Mathematical Solution of
the Enigma Cipher."

"Part One: The Machine": The Enigma cipher machine
could be operated in a variety of ways. The rotors were set in the basic
position. Each message had its independently selected key where three letters
were enciphered twice. This way, six letters were obtained. Encipherment
began from the same position in the message, which was unknown to the cryptologist.
The rotors were then set to a selected individual key. Each key was enciphered
twice. If there is a sufficient number of messages in a given day, then
all the letters in the alphabet will occur in all six places at the openings
of the messages. Permutations are formed, thus constituting the starting
point for solving Enigma. These permutations, designated respectively by
the letters A through F, form products AD, BE, and CF because the transitions
by the rotors change letters from the first letter to the fourth letter,
from the second letter to the fifth letter, and from the third letter to
the sixth letter. This is using a specific permutaion. "They may be represented
as disjunctive products of cycles and then assume a very characteristic
form, generally different for each day, for example:

AD = (dvpfkxgzyo) (eijmunqlht) (bc) (rw) (a) (s)

BE = (blfqveoum) (hjpswizrn) (axt) (cgy) (d) (k)

CF = (abviktjgfcqny) (duzrehlxwpsmo)."

Using these sets from days worth of retrieving messages,
the mathematicians managed to reconstruct the machine's internal connections.
According to Rejewski, we know from the machine's description that, if
striking a given key "x" causes the "y" lamp to light, then, conversely,
striking the "y" key will cause the "x" lamp to light. For example, if
the encipherer continues striking keys, and first he strikes the unknown
key "x" and obtains the letter "a", and in the fourth letter place, he
again strikes the same unknown letter "x" and this time, obtains a "b,"
then by striking the "a" key in the first place, he would obtain the letter
"x," and in the fourth place if he struck the "b" key, he would obtain
the letter "x" as well. There is a successive action occurring - first
of "a" on "x," and then of "x" on "b." So, when "ab" are written next to
each other, a fragment permutaion of AD is written, which is a product
of the unknown permutations A and D.

"Let us now consider the following example. Let

dmq vbn

von puy

puc fmq

designate the openings, that is, the double enciphered
keys, of three of some eighty messages available for a given day." Looking
at the first and fourth letters, notice that "d" becomes "v," "v" becomes
"p," "p" becomes "f." A fraction of the permutation AD, "dvpf" is obtained.
Noticing the second the second and fifth letters, "o" becomes "u," "u"
becomes "m," "m" becomes "b." A fragment of BE, "oumb" is now obtained.
Lastly, from the third and sixth letters, "c" becomes "q," "q" becomes
"n," "n" becomes "y." "cqny," a fragment of CF is also arrived at. "The
openings of further messages would permit a complete assembly of the set
of permutations AD, BE, CF. Because of its configuration and fundamental
importance, we shall call this set the characteristic set, or, simply,
the characteristic for the given day."

After a key has been depressed and before the current
causes a given lamp to light, it starts passing through a series of the
machine's components. Each component causes a permutation of the alphabet.

"If we designate the permutation
caused by the commutator with the letter S, and that which is caused by
the three rotors respectively (from left to right) with the letters L,M,N
and that which is caused by the reversing drum with the letter R, then
the path of the current will be represented by the product of permutations
SNMLRL-1 M-1N-1S-1" [Rejewski]. When the key is pressed, the rotor N turns
one twenty-sixth of a revolution, and Rejewski introduces a special permutation
to name this cycle that transforms each letter of the alphabet into the
next one. He designates it with the letter P:

P = (a b c d e f g h i j k l m n o p q r s t u v
w x y z).

In the diagram,
the current before and after the movement of rotor N is shown. Rejewski
explains that the equations,

A = SPNP-1MLRL-1M-1PN-1P-1S-1

B = SP2NP-2MLRL-1M-1P2N-1P-2S-1

.................................................

E = SP5NP-5MLRL-1M-1P5N-1P-5S-1

F = SP6NP-6MLRL-1M-1P6N-1P-6S-1

make up the known products AD, BE, CF, which are
expressed in the form:

AD = SPNP-1MLRL-1M-1PN-1P3NP-4MLRL-1M-1P4N-1P-4S-1

BE = SP2NP-2MLRL-1M-1P2N-1P3NP-5MLRL-1M-1P5N-1P-5S-1

CF = SP3NP-3MLRL-1M-1P3N-1P3NP-6MLRL-1M-1P6N-1P-6S-1.

The first part of the task is to solve the set of
equations. On the right side only the permutation P and its powers are
known. The permutations S,L,M,N,R are unknown. We now need to simplify
the set. The first step consists in replacing the repeated product MLRL-1M-1,
let's call this the "fictitious" reversing drum, with the single letter
Q. So now the number of unknowns are namely three, S, N, Q:

AD = SPNP-1QPN-1P3NP-4QP4N-1P-4S-1

BE = SP2NP-2QP2N-1P3NP-5QP5N-1P-5S-1

CF = SP3NP-3QP3N-1P3NP-6QP6N-1P-6S-1.

Rejewski also continues to explain the next step,
which includes the Theorem on the Product of Transpositions. From the now
known products AD, BE, CF we need to now determine separately the permutations
A through F. The permutations consist only of transpositions, and AD, BE,
CF are their products. Applying them to the following theorem: If two
permutations of the same degree consist solely of disjunctive transpositions,
then their product will include disjunctive cycles of the same lengths
in even numbers, gets us to Rejewski's proof of the theorem.

"Proof: As an example, we designate the permutations
to be multiplied by each other by the letters X and Y, and their degree
by 2n. If, in permutation X, there happens to occur a transposition identical
with a transposition occurring in permutation Y, for example (ab), then,
in the product XY, there will occur a pair of uniliteral cycles (a) (b).
With reference to transpositions identical in both permutations the theorem
is, therefore, true. After rejecting these identical transpositions, we
may without prejudice to generality assume that

in permutation X in permutation Y

there will occur there will occur

the transposition the transposition

(a1a2) (a2a3)

(a3a4) (a4a5)

..................... .....................

(a2k-3a2k-2) (a2k-2a2k-1)

(a2k-1a2k) (a2ka1)

because the initial letter a1 must finally occur
in permutation Y. When we proceed to execute the multiplication XY, obviously
we shall always obtain two cycles of the same length k ¾ n:

(a1a3.....a2k-3a2k-1) (a2ka2k-2.....a2ka1)."

Not all the letters have been exhausted in this method,
but the procedure is continued until they have all been found. Rejewski
notes that the letters which are in the same transposition are always involved
with two different cycles of the same length, of permutation XY. He also
states that if two letters occur in two different cycles, of the same length,
of permutation XY, which also happen to belong to the same transposition,
then the letters adjacent to them also belong to a single transposition.
The converse theorem states: If a permutation of even-numbered degree
includes disjunctive cycles of the same lengths in even numbers, then this
permutation may be regarded as a product of two permutations, each consisting
solely of disjunctive transpositions.

A proof is unnecessary for this converse theorem
to give a formula for the number of possible solutions for X and Y. When
this theorem is applied to the products AD, BE, CF, supplies for each of
the expressions A,B,C, whereas permutations D,E,F are determined uniquely
by the former. There are several thousands of different possible solutions
for the whole characteristic set of three equations. If we were to extract
the one true solution out of the many that exist, it would be very difficult.
This theorem did not bring us to the goal we want, but it did bring us
closer. If we had some knowledge of the keys for the messages, or some
other knowledge of the encipherers' habits, plus the theorem on the product
of transpositions, this now enables us to find the one correct solution,
so that in the set of equations

A = SPNP-1QPN-1P-1S-1

B = SP2NP-2QP2N-1P-2S-1

...................................

F = SP6NP-6QP6N-1P-6S-1,

we may label the left sides as known. But, most often,
before breaking the cipher, the cryptologist does not usually know the
encipherers' habits.

"Whether the foregoing set of six permutational equations
with three unknowns S, N, Q is soluble without further supplementary data
is not known to this day. However, it is known that this set would be soluble
if the cryptologist had cipher material for two different days, with different
plug connections, but with the same or nearly the same setting of rotors."
The number of possible settings of the rotors is 6 (26) (26) (26) = 105,456.
In a number of days, let us say five hundred, the same setting of the rotors
may appear twice. But, the goal is still tough to reach, because we would
have to go through all the other days as well. In theory, though, it is
still soluble. In reality, though, the necessary supplementary data were
obtained by a much shorter way.

The French Cipher Bureau supplied the Polish Cipher
Bureau with intelligence material containing German tables of Enigma keys,
which also included the S commutator connections, in December 1932. The
S permutation became possible to be transferred, as a known, to the left
side of our set:

S-1AS = PNP-1QPN-1P-1

S-1BS = P2NP-2QPN-1P-2

..................................

S-1ES = P5NP-5QPN-1P-5

S-1FS = P6NP-6QPN-1P-6

having six equations with only two unknowns, N and
Q. "This set is now soluble, but for various reasons, mainly in order to
make certain whether within the six permutations A through F there does
not occur a movement of rotor M, it is advisable to carry out certain transformations."

Transforming both sides of the first equation by
P, of the second equation by P2, and so on, to simplify things, we denote
the left sides by the letters U through Z:

U = P-1S-1ASP = NP-1QPN-1

V = P-2S-1BSP2 = NP-2QP2N-1

.........................................

Y = P-5S-1ESP5 = NP-5QP5N-1

Z = P-6S-1FSP6 = NP-6QP6N-1.

Two products of each two successive expressions are
formed:

UV = NP-1 (QP-1QP) PN-1

VW = NP-2 (QP-1QP) P2N-1

WX = NP-3 (QP-1QP) P3N-1

XY = NP-4 (QP-1QP) P4N-1

YZ = NP-5 (QP-1QP) P5N-1.

By eliminating the common expression QP-1QP, a set
of four equations with only one unknown NPN-1 is obtained:

VW = NP-1N-1 (UV) NPN-1

WX = NP-1N-1 (VW) NPN-1

XY = NP-1N-1 (WX) NPN-1

YZ = NP-1N-1 (XY) NPN-1.

Continuing using this method, from the first equation,
NPN-1, there are several dozen expressions arrived at, depending on the
permutation of UV. The same number of solutions will be obtained for NPN-1
from the second equation, and one of these solutions must be identical
with one solution for the first equation. The final two equations are now
superfluous. We will need to apply the indicated method once again by comparing
the solution obtained for NPN-1 with permutation P. "We will obtain twenty-six
possible solutions for N-1 that do not differ essentially, and, after selecting
one of them, we will readily obtain N itself, the internal connections
of the right-hand rotor."

Breaking Enigma

The mathematical solution of the Enigma cipher was put
together by Marian Rejewski. The following information appears in Appendix
E from Enigma: How the German Machine Cipher Was Broken, and How It
Was Read by the Allies in World War Two. It is a primary source by
Marian Rejewski, himself, which reports on "The Mathematical Solution of
the Enigma Cipher."

"Part Two: Keys": Methods had to be devised for rapid
reconstruction of the daily keys in order to solve codes. The permutation
theory will help explain, how, once having the machine, one can reconstruct
the keys. "As the formulas for AD, BE, CF show, permutation S, as a transforming
permutation, influences solely the letters within cycles comprising permutations
AD, BE, CF, but does not influence the actual configuration of these cycles."
There are three rotors which can be placed on an axis in six different
ways. The rotors can take on (26) (26) (26) = 17,576 different positions,
as described earlier. If a machine existed, which could give the length
and number of cycles, and if these figures were catalogued, then the products
AD, BE, CF could be compared for a given day with products of the same
configuration. Such a devise was designed, and its name was the cyclometer.

The main part of the cyclometer
comprised two sets of rotors suitably connected by wires through which
electric current could run. The device would turn the rotors one by one
while counting the lamps that light. From this information, one can determine
the length and number of cycles. Since there were six possible sequences,
the catalog of characteristics enveloped a total of (6) (17,576) = 105,
456 entries.

Beginning on September 15, 1938, the catalog of characteristics
based on the cyclometer no longer fulfilled its task. The Germans instilled
completely new rules for enciphering message keys. Hence, the Enigma operator
himself was the one who selected the basic position, a new one each time,
for enciphering the individual message key. The key, as before, was enciphered
twice. The first letter continued to designate the same thing as the fourth
letter, the second the same as the fifth, and do on. The basic position
was now known to the cryptologist, but unfortunately was different for
each message. Therefore, there were no products AD, BE, CF characteristics
daily, whose configuration could be found in the catalog. "Since the length
of the cycles in products AD, BE, CF was invariable with respect to the
transformations produced by permutation S, the occurrence or nonoccurrence
of constant points in the products was invariable with respect to those
transformations."

A catalog of constant points for all the 17,576 possible
products was now needed. If the constant points, which occurred in the
catalog about forty percent of the time, were transferred to a long tape,
they would have formed a distinct pattern. Then from this first tape, if
transferred to a second tape according to the positions of the constant
points, the task would be to determine where it collided with the constant
points on this second tape. Since this was a very difficult method, Henryk
Zygalski came up with the idea of perforated sheets, letters "a" through
"z."

This was a system of coordinates
in which the abscisses and ordinates marked successive possible positions
of rotors. Cases with constant points were perforated. According to the
diagram, we can see that each constant point had to be perforated as many
as four times. The sheets were superimposed and moved with respect to each
other, in accordance with a strictly defined program, so the number of
visible apertures gradually decreased. If the sufficient quantity of data
was available, then there remained a single aperture, probably corresponding
to the solution. Once the position of the aperture was determined, one
could calculate the order of the rotors, the settings of their rings, and
by comparing the cipher letters with the machine letters, (otherwise known
as permutation S,) one could in other words calculate the entire cipher
key.

The German cipher service constantly introduced difficulties
designed to frustrate those trying to recover the keys. These ploys had
to be dealt with. In November 1937, the reversing drum was exchanged. The
number of connections in the commutator was gradually increased form six
to thirteen pairs. In December 1938, the number of rotors was increased
from three to five. German radio communications nets grew from year to
year. Even though they used the same Enigma devices, different keys were
used. By September 1939, almost all the Cipher Bureau's equipment and most
of its records were destroyed due to the evacuation. But, in July of 1939,
the Polish, the French, and the British cipher bureau representatives met
and decided to make available all its devices and methods for Enigma decryptment
to future war allies. They also agreed to hand over a copy of the German
Enigma cipher machine that had been reconstructed in Poland on the basis
of the theoretical work that was described in "The Mathematical Solution
of the Enigma Cipher," by Marian Rejewski.